| L(s) = 1 | − 2.30·2-s − 0.694·3-s + 3.31·4-s + 3.08·5-s + 1.59·6-s + 3.70·7-s − 3.02·8-s − 2.51·9-s − 7.09·10-s + 11-s − 2.29·12-s − 2.20·13-s − 8.54·14-s − 2.13·15-s + 0.342·16-s − 17-s + 5.80·18-s + 1.82·19-s + 10.2·20-s − 2.57·21-s − 2.30·22-s + 8.54·23-s + 2.09·24-s + 4.48·25-s + 5.08·26-s + 3.83·27-s + 12.2·28-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.400·3-s + 1.65·4-s + 1.37·5-s + 0.653·6-s + 1.40·7-s − 1.06·8-s − 0.839·9-s − 2.24·10-s + 0.301·11-s − 0.663·12-s − 0.612·13-s − 2.28·14-s − 0.552·15-s + 0.0855·16-s − 0.242·17-s + 1.36·18-s + 0.419·19-s + 2.28·20-s − 0.561·21-s − 0.491·22-s + 1.78·23-s + 0.428·24-s + 0.897·25-s + 0.997·26-s + 0.737·27-s + 2.31·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.197960388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.197960388\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 2.30T + 2T^{2} \) |
| 3 | \( 1 + 0.694T + 3T^{2} \) |
| 5 | \( 1 - 3.08T + 5T^{2} \) |
| 7 | \( 1 - 3.70T + 7T^{2} \) |
| 13 | \( 1 + 2.20T + 13T^{2} \) |
| 19 | \( 1 - 1.82T + 19T^{2} \) |
| 23 | \( 1 - 8.54T + 23T^{2} \) |
| 29 | \( 1 + 4.48T + 29T^{2} \) |
| 31 | \( 1 + 3.61T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 2.53T + 41T^{2} \) |
| 47 | \( 1 + 3.78T + 47T^{2} \) |
| 53 | \( 1 + 4.35T + 53T^{2} \) |
| 59 | \( 1 + 2.26T + 59T^{2} \) |
| 61 | \( 1 + 5.57T + 61T^{2} \) |
| 67 | \( 1 + 4.49T + 67T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 - 6.97T + 73T^{2} \) |
| 79 | \( 1 - 7.85T + 79T^{2} \) |
| 83 | \( 1 - 0.718T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 5.92T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.80378263440637306002163337820, −7.46180823828350978401643196165, −6.52718145152032466986147886255, −5.93894488477823474123465013267, −5.14718423779579851991437740561, −4.65939262122022206339020851010, −3.01022247659253253867654928589, −2.20294539818112852677596314396, −1.57801522917659272476855235415, −0.74629906286034119426959885139,
0.74629906286034119426959885139, 1.57801522917659272476855235415, 2.20294539818112852677596314396, 3.01022247659253253867654928589, 4.65939262122022206339020851010, 5.14718423779579851991437740561, 5.93894488477823474123465013267, 6.52718145152032466986147886255, 7.46180823828350978401643196165, 7.80378263440637306002163337820
